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arXiv:1903.12027

Born's Rule from Quantum Frequentism

Lionel Brits

Published Mar 28, 2019• 0 citations

Abstract

Quantum theory has evolved from a set of provisional rules to an indispensable framework that underlies much of modern technology and infrastructure. Yet, after a century, Born's probability postulate remains at odds with the theory's unitary character. The problem stems from the linearity of the Schrödinger equation, as linear systems are insensitive to the magnitudes of their solutions' coefficients. If measurement is unitary, and thus linear, how can the frequency of an outcome depend on the magnitude of its amplitude? And if not, at what scale does unitarity break down? This question remains pressing, as the assumption of unitarity underlies both the design of large-scale fault-tolerant quantum devices, as well as our understanding of fundamental aspects of our universe, for example, the black hole information problem. Proponents of the many-worlds interpretation have argued that Born's rule is observed because events that violate it have vanishing norms, and so must be unphysical. However, this argument has only been made explicit for special cases involving infinitely many identical states. In this paper we provide a generalized form of Born's rule applicable to measurements of arbitrarily-prepared factorizable states, and prove that such systems contain no histories that violate Born's rule. Our result therefore demonstrates that purely unitary evolution can co-exist with Born's rule under more relaxed conditions. In addition, we apply our generalized rule to the problem of quantum circuit fidelity, and provide a single-shot alternative to cross-entropy benchmarking based on self-information rate.

DOI: 10.48550/arxiv.1903.12027

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Quantum theory has evolved from a set of provisional rules to an indispensable framework that underlies much of modern technology and infrastructure. Yet, after a century, Born's probability postulate remains at odds with the theory's unitary character. The problem stems from the linearity of the Schrödinger equation, as linear systems are insensitive to the magnitudes of their solutions' coefficients. If measurement is unitary, and thus linear, how can the frequency of an outcome depend on the magnitude of its amplitude? And if not, at what scale does unitarity break down? This question remains pressing, as the assumption of unitarity underlies both the design of large-scale fault-tolerant quantum devices, as well as our understanding of fundamental aspects of our universe, for example, the black hole information problem. Proponents of the many-worlds interpretation have argued that Born's rule is observed because events that violate it have vanishing norms, and so must be unphysical. However, this argument has only been made explicit for special cases involving infinitely many identical states. In this paper we provide a generalized form of Born's rule applicable to measurements of arbitrarily-prepared factorizable states, and prove that such systems contain no histories that violate Born's rule. Our result therefore demonstrates that purely unitary evolution can co-exist with Born's rule under more relaxed conditions. In addition, we apply our generalized rule to the problem of quantum circuit fidelity, and provide a single-shot alternative to cross-entropy benchmarking based on self-information rate.

In order to make the quantum mechanics a closed theory one has to derive the Born rule from the first principles, like the Schroedinger equation (SEq), rather than postulate it. The Born rule was in certain sense derived in several articles, e.g. in [1] and [2]. In this work some arguments of previous authors are simplified and made more “physical”. It is shown how to derive the Born rule using the conservation of quantum state norm (Ψ|Ψ) that is the unitary evolution property determined by the SEq. It is this property that makes the probability equal to the square of the amplitude modulus. Possible modification of the Born rule is briefly discussed in the case when the evolution of the system is described by a nonlinear Schroedinger equation. We also present arguments in the spirit of the Many-World Interpretation to explain the origin of probabilistic behavior. Simply speaking, the randomness appears as a result of representing the wave function by using a detector of discrete nature that is found only in one state at a time, out of two or more possible states.

The Born rule for computing probabilities of the outcomes of measurements is an indispensable ingredient of quantum mechanics. The standard textbook description of this rule gives the impression that it implies the unitarity of time evolution. This view relies on the argument that unless the dynamics is unitary, the probabilities of finding all possible outcomes of a measurement do not add up to 1, i.e., the total probability is not conserved. We show that this argument is flawed, and that the general expression for the Born rule ensures the conservation of total probabilities even when the dynamics of a quantum system is not unitary. This applies to the dynamics of ensembles of quantum systems in both pure and mixed states. We discuss the status of the local conservation of probabilities and the arguments against the plausibility of non-unitary time evolutions that are based on the identification of the Hamiltonian operator with the energy observable.

The inability of Schrödinger's unitary time evolution to describe the measurement of a quantum state remains a central foundational problem. It was recently suggested that the unitarity of Schrödinger dynamics can be spontaneously broken, resulting in measurement as an emergent phenomenon in the thermodynamic limit. Here, we introduce a family of models for spontaneous unitarity violation that apply to generic initial superpositions over arbitrarily many states, using either single or multiple state-independent stochastic components. Crucially, we show that Born's probability rule emerges spontaneously in all cases.

The predictions of quantum mechanics are probabilistic. Quantum probabilities are extracted using a postulate of the theory called the Born rule, the status of which is central to the problem of quantum mechanics. Efforts to justify the Born rule from other physical principles, and thus elucidate the measurement process, have involved lengthy statistical or information-theoretic arguments. Here we show that Bohm's deterministic formulation of quantum mechanics allows the Born rule for measurements on a single system to be derived, without any statistical assumptions. We solve a simple example where the creation of an ensemble of identical quantum states, together with position measurements on those states, are described by Bohm's quantum dynamics. The calculated measurement outcomes agree with the Born-rule probabilities, which are thus a consequence of deterministic evolution. Our results demonstrate that quantum probabilities can emerge from simple dynamical laws alone, and they support the view that there is no underlying indeterminism in quantum phenomena.

The measurement problem remains one of the most profound puzzles in the foundation of quantum mechanics. It contains two distinct aspects: the collapse of the wavefunction and the origin of probabilities. While Everettian Quantum Mechanics (EQM) and the Many Worlds Interpretation (MWI) work around the collapse aspect, by positing that the universe branches through pure unitary evolution, it leaves unresolved the question of why measurement outcomes occur with frequencies given by the Born Rule. This paper explores a recent derivation of the Born Rule by Carroll and Sebens based on principles of Self-Locating Uncertainty (SLU) and the Epistemic Separability Principle (ESP). By examining this framework in detail, beginning with an equal amplitude case and extending to an unequal amplitude case, we show how these principles account for the probabilities given by the Born Rule, hence showing that SLU and the ESP, within the scope of EQM, offer a solution to the measurement problem. We further show the significance and implications of deriving the Born Rule and highlight the relevance of the measurement problem to more recent developments like quantum gravity.

The Born rule stands as the fundamental probability postulate of quantum mechanics, yet its first-principle derivation remains a longstanding open problem. This paper presents a rigorous, collapse-free, and observer-independent derivation of the Born rule. Starting from four physically natural and minimal axioms—unitary dynamics, uniqueness of classical records, unitary invariance, and consistency with the classical limit and ergodicity—we uniquely deduce the Born rule from first principles. We prove that the Born rule is not an independent postulate, but a necessary mathematical consequence of the core axiomatic structure. Our approach is self-contained, assumption-free, and purely unitary, establishing that the Born rule is canonically determined by the fundamental mathematical and physical principles of quantum mechanics.